# CA ADV Algebra Standard 07

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```					CA ADV Algebra Standard 07

Multiple Choice
Identify the choice that best completes the statement or answers the question.

____    1. Simplify             . Identify any z-values for which the expression is undefined.
a. 2z; no excluded values
b. 2z; z 3 or 0
c. 2z; z 3
d.             ; z 3 or 0

____    2. Simplify               . Identify any x-values for which the expression is undefined.
a.

The expression is undefined at        .
b.

The expression is undefined at       and         .
c.

The expression is undefined at       and         .
d.

The expression is undefined at        .

____    3. Multiply                   . Assume that all expressions are defined.

a.
b.
c.
d.

____    4. Divide               . Assume that all expressions are defined.

a.

b.

c.

d.
a.
b.
c.
d. There is no solution because the original equation is undefined at       .

____   6. Subtract                         . Identify any x-values for which the expression is undefined.
a.
; The expression is always defined.
b.
; The expression is undefined at x = ±3.
c.
; The expression is undefined at x = ±3.
d.
; The expression is always defined.

a.

b.

c.

d.

____   8. Subtract                  . Identify any x-values for which the expression is undefined.
a.

The expression is undefined at       and          .
b.

The expression is undefined at       and          .
c.

The expression is undefined at       and          .
d.

The expression is undefined at       and          .
____   9. Simplify                        . Assume that all expressions are defined.

a.

b.

c.

d.

____ 10. Solve the equation                      .
a.
b.         or
c.          or
d.

____ 11. Solve the equation                          .
a. There is no solution.
b.      3
2
c.
d.

____ 12. Solve the inequality                    algebraically.
a.             or x >  6
13

b.  13
6
c.         6 or
13

d.              6
13

____ 13. Solve the equation                                       .
a.         or
b.         ,           , or
c.         or
d.         or

MULTIPLE CHOICE

1. ANS: B
Factor common factors out of the numerator and/or denominator. Divide out the common factors to simplify
the expression. Finally, use the original denominator to determine any z-values for which the expression is
undefined.

Feedback
A     Determine excluded values from the original denominator.
B     Correct!
C     Determine excluded values from the original denominator.
D     Divide out common factors.

PTS:    1              DIF: Average         REF: Page 577
OBJ:    8-2.1 Simplifying Rational Expressions                 NAT: 12.5.3.c
STA:    2A7.0          TOP: 8-2 Multiplying and Dividing Rational Expressions
2. ANS:    B

Factor       from the numerator and reorder the terms.

=                                  Factor the numerator and denominator.

=                                  Divide the common factors and simplify.

The expression is undefined at those x-values, 2 and        , that make the original denominator 0.

Feedback
A     Look at the original expression to find the values that make it undefined.
B     Correct!
C     Don't forget to redistribute the –1.
D     Don't forget to redistribute the –1. Look at the original expression to find the values that
make it undefined.

PTS: 1                 DIF: Average             REF: Page 578      OBJ: 8-2.2 Simplifying by Factoring -1
NAT: 12.5.3.c          STA: 2A7.0               TOP: 8-2 Multiplying and Dividing Rational Expressions
3. ANS: C

Arrange the expressions so like terms are together:                          .

Multiply the numerators and denominators, remembering to add exponents when multiplying:               .

Divide, remembering to subtract exponents:              .
Since         , this expression simplifies to       .
Feedback
A    A variable raised to the 0 power simplifies to 1.
B    When dividing powers with the same base, subtract the exponents.
C    Correct!
D    Multiply, then simplify.

PTS:   1              DIF: Basic           REF: Page 578
OBJ:   8-2.3 Multiplying Rational Expressions                 NAT: 12.5.3.c
STA:   2A7.0          TOP: 8-2 Multiplying and Dividing Rational Expressions
4. ANS:   A

Rewrite as multiplication by the reciprocal.

=                           Simplify by canceling common factors.

=

Feedback
A    Correct!
B    To divide by a fraction, you multiply by its reciprocal.
C    To divide by a fraction, you multiply by its reciprocal.
D    Multiply the first fraction by the reciprocal of the second fraction.

PTS:   1              DIF: Basic           REF: Page 579
OBJ:   8-2.4 Dividing Rational Expressions                    NAT: 12.5.3.c
STA:   2A7.0          TOP: 8-2 Multiplying and Dividing Rational Expressions
5. ANS:   D
Note that       .

Factor.
The factor          cancels.

Because the left side of the original equation is undefined when        , there is no solution.

Feedback
A    Is the original equation defined for this value of x?
B    Factor the numerator and cancel common factors before solving for x. Is the original
equation defined for this value of x?
C    Factor the numerator and cancel common factors before solving for x. Is the original
equation defined for this value of x?
D    Correct!

PTS: 1              DIF: Average         REF: Page 579
OBJ: 8-2.5 Solving Simple Rational Equations                         NAT: 12.5.3.c
STA: 2A7.0            TOP: 8-2 Multiplying and Dividing Rational Expressions
6. ANS: D

=                                 Subtract the numerators. Distribute the negative sign.

=                                 Combine like terms.

There is no real value of x for which x2 9 = 0; the expression is always defined.

Feedback
A    Did you add or subtract the numerators?
B    Did you add or subtract the numerators?
C    The expression is undefined at x = c if and only if the denominator is zero at x = c.
D    Correct!

PTS:   1             DIF: Basic           REF: Page 583
OBJ:   8-3.1 Adding and Subtracting Rational Expressions with Like Denominators
NAT:   12.5.3.c      STA: 2A7.0           TOP: 8-3 Adding and Subtracting Rational Expressions
7. ANS:   D
Factor the denominators. The LCD is                    .

=                                     Multiply by           .

=

=                                     Factor the numerator.

=                                     Divide the common factor.

Feedback
A    Find a common denominator before adding the fractions.
B    This is the first fraction rewritten with the common denominator. Add this to the second
fraction.
C    Find a common denominator before adding the fractions.
D    Correct!

PTS: 1                DIF: Average           REF: Page 584     OBJ: 8-3.3 Adding Rational Expressions
NAT: 12.5.3.c         STA: 2A7.0             TOP: 8-3 Adding and Subtracting Rational Expressions
8. ANS: A
Factor the denominators.

The LCD is                 , so multiply        by
=
.

=                                    Subtract the numerators.

=                                    Multiply the binomials in the numerator.

=                                    Distribute the negative sign.

=                                    Write the numerator in standard form.

=                                    Factor the numerator, and divide out common factors.

The expression is undefined at      and         because these values of x make the factors       and
equal 0.

Feedback
A    Correct!
B    Check your distribution of the negative sign.
C    Did you factor the numerator and divide out common factors correctly?
D    Did you factor the numerator and divide out common factors correctly?

PTS: 1              DIF: Average          REF: Page 585
OBJ: 8-3.4 Subtracting Rational Expressions                  NAT: 12.5.3.c
STA: 2A7.0          TOP: 8-3 Adding and Subtracting Rational Expressions
9. ANS: C
Method 1 Write the complex fraction as division.

Divide.

=                                             Multiply by the reciprocal.

=                                             Add by finding the LCD.

=                                             The common factor (       ) cancels.

=                                             Simplify.

Method 2 Multiply the numerator and denominator of the complex fraction by the LCD of the fractions in the
numerator and denominator.
The LCD is            .

=                                             Cancel common factors.

=                                             Simplify.

Feedback
A   Multiply both addends in the numerator by the reciprocal of the denominator, not just
the first.
B   Multiply the numerator by the reciprocal of the denominator.
C   Correct!
D   Multiply the numerator by the reciprocal of the denominator.

PTS:   1              DIF: Average        REF: Page 586
OBJ:   8-3.5 Simplifying Complex Fractions                     NAT: 12.5.3.c
STA:   2A7.0          TOP: 8-3 Adding and Subtracting Rational Expressions
10. ANS:   B
Multiply each term by the LCD.
Simplify. Note
Write in standard form.
Factor.
or                 Apply the Zero-Product Property.
or                      Solve for x.

Check:

Feedback
A   Multiply each term by the LCD. Then solve the resulting quadratic equation. There may
be more than one solution.
B   Correct!
C   Multiply each term by the LCD. Then solve the resulting quadratic equation.
D   Multiply each term by the LCD. Then solve the resulting quadratic equation. There may
be more than one solution.

PTS: 1               DIF: Average         REF: Page 600          OBJ: 8-5.1 Solving Rational Equations
NAT: 12.5.4.a          STA: 2A7.0             TOP: 8-5 Solving Rational Equations and Inequalities
11. ANS: A
Multiply each term by the LCD, (x – 3).
Simplify. Note that x g 3.
Solve for x.

The solution       is extraneous because it makes the denominators of the original equation equal to 0.
Therefore the equation has no solution.

Feedback
A    Correct!

PTS: 1                DIF: Average            REF: Page 601      OBJ: 8-5.2 Extraneous Solutions
NAT: 12.5.4.a         STA: 2A7.0              TOP: 8-5 Solving Rational Equations and Inequalities
12. ANS: A
Case 1: LCD is positive.                          Case 2: LCD is negative.
Step 1 Solve for x.                               Step 1 Solve for x.

6                                                6
13                                                13

Step 2 Consider the value of the LCD.             Step 2 Consider the value of the LCD.

For case 1, the solution must satisfy the         For case 2, the solution must satisfy the
 6 and                                            6 and
13                                                 13
compound inequality                               compound inequality
6 .                     which simplifies to        .
13
which simplifies to

6 .
13
The solution to the original inequality is the union of the solutions to the two cases:       or

Feedback
A    Correct!
B    Solve for the case when the LCD is positive and the case when the LCD is negative.
D    Solve for the case when the LCD is positive and the case when the LCD is negative.

PTS: 1              DIF: Average          REF: Page 604
OBJ: 8-5.6 Solving Rational Inequalities Algebraically       NAT: 12.5.4.a
STA: 2A7.0          TOP: 8-5 Solving Rational Equations and Inequalities
13. ANS: D

Factor the denominator.
Multiply each term by the LCD
and simplify. Note that        ,       , and      .
Use the Distributive Property.
Write in standard form.
Factor.
or                              Use the Zero-Product Property.
or                               Solve for x.

Feedback
A   Check your answer. Substitute –13 for x in the original equation to see if you get a true
statement.
B   A rational expression is undefined for any value of a variable that makes a denominator
in the expression equal to 0.
C   First, factor the denominator, and then multiply each term by the LCD. Then, factor and
solve for x.
D   Correct!

PTS: 1             DIF: Advanced         NAT: 12.5.4.a           STA: 2A7.0
TOP: 8-5 Solving Rational Equations and Inequalities

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